Content text 2.RELATIONS AND FUNCTIONS.pdf
2. RELATIONS AND FUNCTIONS (1.) Consider the following relations : R x y x y ={ , , ( )∣ are real numbers and x wy = for some rational number w} ; , , , m p S m n p n q = ∣ and q are integers such that n q, 0 and qm pn = } . Then [AIEEE-2010] (a.) R is an equivalence relation but S is not an equivalence relation (b.) Neither R nor S is an equivalence relation (c.) S is an equivalence relation but R is not an equivalence relation (d.) R and S both are equivalence relations (2.) If ( ) 2 2 R x y x y Z x y = + , ; , , 3 8 is a relation on the set of integers Z , then the domain of 1 R − is [JEE (Main)-2020] (a.) 0,1 (b.) − − 2, 1,1, 2 (c.) −1,0,1 (d.) − − 2, 1,0,1, 2 (3.) Let R1 and R2 be two relation defined as follows : ( ) 2 2 2 1 R a b R a b Q = + , : and ( ) 2 2 2 2 R a b R a b Q = + , : , where Q is the set of all rational numbers. Then [JEE (Main)-2020] (a.) Neither R1 nor R2 is transitive. (b.) R2 is transitive but R1 is not transitive. (c.) R1 and R2 are both transitive. (d.) R1 is transitive but R2 is not transitive. (4.) Let f R R : → be defined as f x x ( ) = − 2 1 and g R R : 1 − → be defined as ( ) 1 2 1 x g x x − = − . Then the composition function f g x ( ( )) is : [JEE (Main)-2021] (a.) neither one-one nor onto (b.) onto but not one-one (c.) both one-one and onto (d.) one-one but not onto (5.) Let R P Q P = { , ( )∣ and Q are at the same distance from the origin } be a relation, then the equivalence class of (1, 1− ) is the set : [JEE (Main)-2021] (a.) ( ) 2 2 S x y x y = + = , 2 ∣ (b.) ( ) 2 2 S x y x y = + = , 1 ∣ (c.) ( ) 2 2 S x y x y = + = , 2 ∣ (d.) ( ) 2 2 S x y x y = + = , 4 ∣
(6.) Let A = 2,3, 4,5, ,30 and ' ' be an equivalence relation on A A , defined by (a b c d , , ) ( ) , if and only if ad bc = . Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4,3) is equal to : [JEE (Main)-2021] (a.) 7 (b.) 8 (c.) 5 (d.) 6 (7.) Let N be the set of natural numbers and a relation R on N be defined by ( ) 3 2 2 3 R , N N : 3 3 0 . = − − + = x y x x y xy y Then the relation R is [JEE (Main)- 2021] (a.) An equivalence relation (b.) Reflexive and symmetric, but not transitive (c.) Reflexive but neither symmetric nor transitive (d.) Symmetric but neither reflexive nor transitive (8.) Let Z be the set of all integers, ( ) 2 2 A x y x y = − + , : ( 2) 4 , Z Z ( ) 2 2 B x y x y = + , : 4 and Z Z ( ) ( ) 2 2 2 C x y x y = − + − , : ( 2) 2 4 Z Z If the total number of relations from A B to A C is 2 p , then the value of p is [JEE (Main)-2021] (a.) 16 (b.) 49 (c.) 25 (d.) 9 (9.) Which of the following is not correct for relation R on the set of real numbers? [JEE (Main)-2021] (a.) ( x y R x y , 1 ) − is reflexive and symmetric. (b.) ( x y R x y , 0 1 ) − is neither transitive nor symmetric (c.) ( x y R x y , 0 1 ) − is symmetric and transitive (d.) ( x y R x y , 1 ) − is reflexive but not symmetric (10.) Let R a b a b 1 = − ( , : 13 ) N N and R a b a b 2 = − ( , : *13 ) N N . Then on N: [JEE (Main)-2022] (a.) Both R1 and R2 are equivalence relations (b.) Neither R1 nor R2 is an equivalence relation (c.) R1 is an equivalence relation but R2 is not (d.) R2 is an equivalence relation but R1 is not (11.) Let a set A A A A = 1 2 k , where A A i j = for i j i j k ,1 , . Define the relation R from A to A by R x y y A = ( , :) i if and only if 1b x A i k). Then, R is : [JEE (Main)-2022] (a.) reflexive, symmetric but not transitive (b.) reflexive, transitive but not symmetric
(c.) reflexive but not symmetric and transitive (d.) an equivalence relation (12.) Let R1 and R2 be two relations defined on R by a R1 b ab 0 and 2 aR b a b . Then, [JEE (Main)-2022] (a.) R1 is an equivalence relation but not R2 (b.) R2 is an equivalence relation but not R1 (c.) Both R1 and R2 are equivalence relations (d.) Neither R1 nor R2 is an equivalence relation (13.) For N , consider a relation R on N given by R x y x y = + { , : 3 ( ) is a multiple of 7 } . The relation R is an equivalence relation if and only if [JEE (Main)-2022] (a.) =14 (b.) is a multiple of 4 (c.) 4 is the remainder when is divided by 10 (d.) 4 is the remainder when is divided by 7 (14.) Let R be a relation from the set 1, 2,3, ..,60 to itself such that R a b b pq = = { , : ( ) , where p q, 3 are prime numbers}. Then, the number of elements in R is : [JEE (Main)-2022] (a.) 600 (b.) 660 (c.) 540 (d.) 720 (15.) The relation R a,b : gcd a,b 1,2a b,a,b = = ( ) ( ) Z is:4 Jan 2023(Evening)] (a.) transitive but not reflexive (b.) symmetric but not transitive (c.) reflexive but not symmetric (d.) neither symmetric nor transitive (16.) The equation 2 x x x x x − + + = 4 3 , where x denotes the greatest integer function, has: [24 Jan 2023(Evening)] (a.) exactly two solutions in (− , ) (b.) no solution (c.) a unique solution in (−,1) (d.) a unique solution in (− , ) (17.) Let R be a relation on N N defined by (a, b R) (c d, ) if and only if ad b c bc a d ( − = − ) ( ) . Then R is [31 Jan 2023(Morning)] (a.) symmetric but neither reflexive nor transitive (b.) transitive but neither reflexive nor symmetric (c.) reflexive and symmetric but not transitive (d.) symmetric and transitive but not reflexive
(18.) Let P S( ) denote the power set of S = 1, 2,3, ,10 . Define the relations R1 and R2 on P S( ) as AR B1 if ( ) ( ) c c A B B A = and AB2 if c A B = ( ) c B A , A, B P S . Then : [01 Feb 2023(Evening)] (a.) both R1 and R2 are equivalence relations (b.) only R1 is an equivalence relation (c.) only R2 is an equivalence relation (d.) both R1 and R2 are not equivalence relations (19.) The minimum number of elements that must be added to the relation R a,b , b,c = ( ) ( ) on the set abc , , so that it becomes symmetric and transitive is:[30 Jan 2023(Morning)] (a.) 4 (b.) 7 (c.) 5 (d.) 3 (20.) Let R be a relation defined on N as a b R is 2 3 a b + is a multiple of 5, , a bN. Then R is [29 Jan 2023(Evening)] (a.) not reflexive (b.) transitive but not symmetric (c.) symmetric but not transitive (d.) an equivalence relation
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